Although the weight matrix is the same and the solution of the Riemann–Hilbert problem is unique, if we perform the transformation (REF ) with \({T}{S}\)  instead of \({T}\)  then we should expect a new Lax pair with additional information. This was already pointed out in [1]}, where the authors applied this approach to simplify considerably the differential relations for some Hermite-type matrix-valued examples. The situation in the scalar case is pointless and it is not possible to get new relations from this approach (see [1]}). Nevertheless in order to produce new interesting relations, we have to take an specific choice of the matrix \({B}\) .
